Gaspari was born in Gissi, in Abruzzo, a region which would be his main electoral basin during his political career. He graduated in law at the University of Bologna.

Gaspari was a member of the Italian parliament without interruptions from 1968 to 1992. He was a member of the so-called Dorotei wing of Italian Christian Democracy (Italian: Democrazia Cristiana, simply DC) along with Vincenzo Scotti, Arnaldo Forlani and Antonio Gava. In 1960 he was undesecretary to Mail and Telecommunications in the Tambroni Cabinet. After a series of positions as undersecretary in various ministers, he became minister for the first time () in the second Rumor government (1968), and subsequently, again under Rumor and the follower prime ministers Colombo and Andreotti, he was Minister of Reformation of Public Administration.

In the Andreotti II Cabinet he was Minister of Health. Gaspari was also national vice-secretary of DC in 1976–1980. In 1980 he became Minister of Relationships with the Parliament in the second Cossiga government, Minister of Public Functions in the two Bettino Craxi-led cabinets, and, from 1987, Minister of Defence in the short-lived Amintore Fanfani government. After a period as Minister for Mezzogiorno, he ended his government career as Minister of Public Functions in the sixth and seventh cabinets led by Giulio Andreotti in the early 1990s.